
# q-expansion of newform 2736.2.a.m, downloaded from the LMFDB on 20 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_data))]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 2736
weight = 2
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [0], [0], [0], [2], [-2], [-4], [-1], [2], [-6], [-4], [2], [2], [-4], [10], [-6], [4], [2], [-4], [4], [-6], [-8], [-2], [-6], [-10], [-4], [0], [12], [-2], [-6], [-12], [-6], [-4], [-12], [20], [8], [2], [4], [-8], [-10], [-12], [-2], [-10], [-6], [8], [0], [-12], [-16], [24], [2], [16], [6], [-18], [-18], [22], [-10], [14], [16], [18], [6], [12], [14], [24], [-30], [-2], [-18], [-8], [-18], [-22], [22], [4], [14], [8], [2], [0], [-36], [-16], [38], [10], [-26], [-6], [22], [12], [2], [16], [18], [-34], [-14], [28], [8], [6], [30], [28], [-30], [-4], [30], [14], [18], [-4], [14], [32], [-12], [16], [-42], [4], [22], [22], [-8], [-16], [2], [12], [26], [-36], [-4], [32], [-30], [-20], [18], [12], [40], [10], [-6], [-30], [36], [36], [-20], [-26], [-30], [32], [-18], [44], [44], [-40], [26], [52], [-10], [-26], [28], [-42], [-48], [-24], [4], [-16], [12], [30], [4], [6], [-46], [-20], [-36], [-22], [52], [44], [24], [-52], [40], [16], [8], [-38], [30], [-38], [6], [32], [0], [18], [-8], [-20], [26], [-10], [-36], [28], [-38], [-46], [-10], [-36], [16], [-52], [60], [44], [14], [-16], [-48], [-50], [14], [-14], [44], [38], [-4], [-2], [2], [-34], [-6], [8], [42], [50], [-6], [2], [-66], [0], [-42], [-6], [4], [-50], [-58], [-54], [-12], [-40], [-36], [36], [-8], [22], [-24], [-48], [-4], [-4], [-46], [-32], [-38], [20], [40], [-66], [-32], [50], [16], [8], [-6], [38], [20], [52], [-62], [-30], [8], [-36], [-14], [-20], [-6], [50], [24], [42], [-36], [-44], [20], [40], [-54], [-18], [-18], [-32], [-24], [-56], [18], [-14], [48], [50], [-26], [-64], [66], [-42], [16], [18], [38], [-64], [72], [-42], [-6], [-72], [-4], [-54], [40], [-64], [0], [-22], [20], [6], [-64], [22], [-48], [-10], [10], [10], [-42], [60], [72], [-2], [10], [20], [-14], [30], [-58], [56], [-42], [-20], [-34], [-42], [36], [-46], [-48], [-64], [48], [0], [84], [70], [-30], [56], [84], [-12], [38], [48], [-74], [74], [-42], [-34], [16], [58], [72], [18], [78], [-14], [0], [26], [-60], [20], [58], [-22], [-16], [-4], [6], [-80], [-28], [12], [-28], [50], [-46], [-88], [42], [60], [-2], [-54], [18], [22], [80], [-62], [-20], [2], [20], [-14], [48], [42], [-36], [-64], [60], [68], [-18], [-8], [60], [-70], [34], [-42], [-42], [-36], [36], [62], [20], [-12], [-84], [68], [2], [68], [52], [-38], [-18], [16], [-22], [-78], [56], [-2], [-32], [42], [34], [32], [-26], [18], [-64], [-8], [-24], [-12], [-68], [-32], [50], [-4], [40], [-50], [34], [18], [68], [-12], [94], [-44], [54], [-92], [16], [-22], [-24], [-40], [22], [-26], [8], [46], [14], [16], [-54], [2], [72], [-48], [68], [58], [60], [48], [100], [52], [28], [82], [-46], [-14], [-64], [50], [-42], [-68], [-24], [-28], [2]]
